I believe that is because English grammar is in a transitional stage right now as we lose declensions for our pronouns. In a century or two, "they" might be the proper genderless pronoun, and we may not even have gendered pronouns at that point -- but that is not the English of today. So while using "they" as a genderless singular pronoun might sometimes work it still feels a bit awkward.
Personally, I try to alternate between "he" and "she" when I am talking about hypothetical people, so I am never pushing the bounds of English grammar (at least with pronouns).
'He' and 'she' switching confuses me, and while in some cases I do feel that it's a result of actual ambiguity, a big part of it might just be unfamiliarity.
Either way, even in Dutch I find myself to be a bit more sensitive to these things and it's not something I'd argue against. Although in Dutch there's not even a 'they' we can use, practically.
Personally, I try to alternate between "he" and "she" when I am talking about hypothetical people, so I am never pushing the bounds of English grammar (at least with pronouns).